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I. In this second volume, we continue at first the study of non homogeneous boundary value problems for particular classes of evolu tion equations. 1 In Chapter 4 , we study parabolic operators by the method of Agranovitch-Vishik [lJ; this is step (i) (Introduction to Volume I, Section 4), i.e. the study of regularity. The next steps: (ii) transposition, (iii) interpolation, are similar in principle to those of Chapter 2, but involve rather considerable additional technical difficulties. In Chapter 5, we study hyperbolic operators or operators well defined in thesense of Petrowski or Schroedinger. Our regularity results (step (i)) seem to be new. Steps (ii) and (iii) are all3.logous to those of the parabolic case, except for certain technical differences. In Chapter 6, the results of Chapter'> 4 and 5 are applied to the study of optimal control problems for systems governed by evolution equations, when the control appears in the boundary conditions (so that non-homogeneous boundary value problems are the basic tool of this theory). Another type of application, to the characterization of "all" well-posed problems for the operators in question, is given in the Ap pendix. Still other applications, for example to numerical analysis, will be given in Volume 3.
Contents
4 Parabolic Evolution Operators. Hilbert Theory.- 1. Notation and Hypotheses. First Regularity Theorem.- 2. The Spaces Hr, s(Q). Trace Theorems. Compatibility Relations.- 3. Evolution Equations and the Laplace Transform.- 4. The Case of Operators Independent of t.- 5. Regularity.- 6. Case of Time-Dependent Operators. Existence of Solutions in the Spaces H2r m, m(Q), Real r ? 1.- Adjoint Isomorphism of Order r.- 8. Transposition of the Adjoint Isomorphism of Order r. (I): Generalities.- 9. Choice of f. The Spaces ?2rm,r(Q).- 10. Trace Theorems for the Spaces D?(r?1)(P)(Q), r ? 1.- 11. Choice of gj and uo. The Spaces H2?m ??(?).- 12. Transposition of the Adjoint Isomorphism of Order ?. (II): Results; Existence of Solutions in H2mr,r(Q)-Spaces, Real r ? 0.- 13. State of the Problem. Complements on the Transposition of the Adjoint Isomorphism of Order 1.- 14. Some Interpolation Theorems.- 15. Final Results; Existence of Solutions in the Spaces H2mr,r(Q), 0 < r < 1. Applications.- 16. Comments.- 17. Problems.- 5 Hyperbolic Evolution Operators, of Petrowski and of Schroedinger. Hilbert Theory.- 1. Application of the Results of Chapter 3 and General Remarks.- 2. A Regularity Theorem (I).- 3. Regular Non-Homogeneous Problems.- 4. Transposition.- 5. Interpolation.- 6. Applications and Examples.- 7. Regularity Theorem (II).- 8. Non-Integer Order Regularity Theorem.- 9. Adjoint Isomorphism of Order r and Transposition.- 10. Choice of f, $$
\vec g
$$, u0, u1.- 11. Trace Theorems in the Space $$
D_{A + D_t^2}^{ - \left( {2r - 1} \right)}\left( Q \right)
$$.- 12. Schroedinger Type Equations.- 6 Applications to Optimal Control Problems.- 1. Statement of the Problems for the Linear Parabolic Case.- 2. Choice of the Norms in the Cost Function.- 3. Optimality Conditionfor Quadratic Cost Functions.- 4. Optimality Condition and Green's Formula.- 5. The Particular Case $$
\mu \,\, = \,\,m\,\, + \,\,\frac{1}{2}
$$, E3 ? 0.- 6. Consequences of the Optimality Condition (I).- 7. Consequences of the Optimality Condition (II).- 8. Complements on the Choice of the Spaces Ki.- 9. Examples.- 10. Non-Parabolic Cases. Statement of the Problems. Generalities.- 11. Applications. Examples.- 12. Comments.- 13. Problems.- Boundary Value Problems and Operator Extensions.- 1. Statement of the Problem. Well-Posed Spaces.- 1.1 Notation.- 2. Abstract Boundary Conditions.- 2.1 Boundary Spaces and Operators.- 2.2 Characterization of Well-Posed Spaces.- 3. Example 1. Elliptic Operators.- 3.1 Notation.- 3.2 The Boundary Operators and Spaces.- 3.3 Consequences.- 3.4 Various Remarks.- 4. Example 2. Parabolic Operators.- 4.1 Notation.- 4.2 The Boundary Operators and Spaces.- 4.3 Consequences.- 5.1 Notation.- 5.2 Formal Results.- 6. Comments and Problems.
